n?u=RePEc:bde:wpaper:1526&r=all

repec.nep.mac

n?u=RePEc:bde:wpaper:1526&r=all

for a constant d. The intuition is simple. The bank/SPV would like to maximize the

volume of securitized assets, but due to the information asymmetry, is forced to retain

some synthetic assets in the portfolio when the signal is good and information asymmetry

is high. Hence, it is optimal for the SPV to select a security that is as payoff insensitive

as possible for the range of signals where asymmetry is at its highest. Standard debt has

this property since f does not change significantly as X increases in the range X ≫ d.

That way, the bank problem is

max E[Π SPV (f d ,f0 d )], where f d = E[min(d, Y n ) | X].

d

In order to provide further characterization on the debt contract, P F and q, we assume

that η ∼ N(0,σ 2 ) (recall that η is the aggregate credit shock affecting all credit assets)

and X n is uniformly distributed between X 0 = ∑ i x i0 and X 1 = ∑ i x i1. As we increase

the number of securities n, thevalueoff becomes

f d n = E[min(d, X n +(1/n) ∑ ɛ i + η) | X] → E[min(d, X n + η) | X] =f d

f d =

∫ (d−X n )

−∞

(X n + η)f(η)dη +

∫ ∞

(d−X n )

df (η)dη

where f(η) is the density function of η

( ) ( ( )) ( )

d − X

f d = X n n

d − X

n

d − X

n

Φ

+ d 1 − Φ

− σφ

σ

σ

σ

where Φ(·) andφ(·) are the standard normal cumulative and density functions

( ) ( ( )) ( )

Also note that f0 d = X d−X0


σ

+ d 1 − Φ d−X0

σ

− σφ d−X0

σ

. The aggregate

shock η, which is not diversified away as the basket of credit is constructed can also be

understood as the correlation risk amongst assets Y i ,fori ∈ [1,n], within the basket.

Based on the solution for f d and f0 d, d∗ is given by

∫ X1

d ∗ =argmax (1 − δ)(f0 d ) 1/(1−δ) (f d ) −δ/(1−δ) 1

dX (4)

X 0

(X 1 − X 0 )

We are not able to obtain an analytical solution to this integral and thus offer a

description of the main trade-off involved in the selection of d ∗ . Due to the presence of

information asymmetry the SPV is forced to retain a fraction (1−q) of synthetic securities.

That is costly since it prevents the SPV from maximizing liquidity creation. Hence, one

of the drivers behind the selection of d ∗ is to minimize the information sensitivity of F .

Figure 6 shows the pay-off of the synthetic security to the final investor for a low value

of d = ˜d (left-hand side) and for a high value of d = ˆd

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